The commutant of fermionic Gaussian unitaries

TL;DR

This paper characterizes the algebraic structure of commutants of fermionic Gaussian unitaries, providing formulas, bases, and connections to fermionic correlations and invariants.

Contribution

It introduces a unified algebraic framework for higher-order invariants of fermionic Gaussian unitaries using Howe dualities and Gelfand--Tsetlin procedures.

Findings

Derived formulas for the dimensions of commutants as functions of t and n.

Constructed explicit orthonormal bases for the commutants.

Connected the algebraic structure to measures of fermionic correlations.

Abstract

In this work, we characterize the $t$-th order commutants of fermionic Gaussian unitaries and of their particle-preserving subgroup acting on $n$ fermionic modes. These commutants govern Haar averages over the corresponding groups and therefore play a central role in fermionic randomized protocols, invariant theory, and resource quantification. Using Howe dualities, we show that the particle-preserving commutant is generated by generalized copy-hopping operators, while that for general Gaussian commutant is generated by generalized quadratic Majorana bilinears together with parity. We then derive closed formulas for the dimensions of both commutants as functions of $t$ and $n$, and develop constructive Gelfand--Tsetlin procedures to obtain explicit orthonormal bases, with detailed low-$t$ examples. Our framework also clarifies the structure of replicated fermionic states and connects naturally to measures of fermionic correlations, generalized Pl\"ucker-type constraints, and the stabilizer entropy of fermionic Gaussian states. These results provide a unified algebraic description of higher-order invariants for fermionic Gaussian dynamics.